Numerical solution of time domain integral equations using the Nyström method

A wide variety of electromagnetic scattering problems are, in principle, most efficiently solved with time domain integral equations. Integral equation methods are well suited for problems involving homogenous scatterers, and time domain methods can efficiently solve broadband, nonlinear, and time-varying problems. While a Galerkin approach is usually used in their solution, recently, the locally corrected Nyström method (Canino, L.F. et al., 1998) was applied to the two-dimensional time domain integral equations (Wildman, R.A. and Weile, D.S., IEEE APS Int. Symp., 2004). The Nyström method has several benefits over Galerkin´s method. In the approach described by Wildman and Weile, a standard Nyström discretization was used in space, and a low-order approximation (trapezoidal rule) was used for the temporal integrals. To obtain accurate results with low-order integration, the kernel of the integral equation was filtered, and bandlimited extrapolation was used to recover a causal representation. This paper follows a similar approach. Fortunately, in three dimensions, the filtering can be performed analytically, so the method is greatly simplified.